As a mathematician, something that I always envied physicists is the uninhibited way they use mathematics.
The classic example is the Dirac delta function, which is a function that’s zero everywhere except the origin, but has area one. The fact that no such function exists is only a minor inconvenience. Delta functions can be made rigorous as a distribution, but the concept well predates its formal definition. For example, Green’s functions, which are defined in terms of the delta function, date from 1828.
A more dramatic example is the concept of a dipole. A dipole is the limit of two electric charges of opposite charge as the distance between them goes to zero. It can also be thought of as the limit of the difference of two Dirac delta functions, or even the derivative of a delta function. Dipoles are used to approximate the effect of magnets from a long distance. In terms of distributions, a dipole is the derivative operator on the space of smooth functions, but this is far from the physical intuition.